Abstract
If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path \(\gamma \) from 0 to infinity such that \(f(z) - a\) tends to 0 as z tends to infinity along \(\gamma \). The Denjoy–Carleman–Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2, mean type. A long-standing problem asks whether this conclusion holds for entire functions having n distinct asymptotic (entire) functions, each of growth at most order 1/2, minimal type. In this paper conditions on the function f and associated asymptotic paths are obtained that are sufficient to guarantee that f satisfies the conclusion of the Denjoy–Carleman–Ahlfors Theorem. In addition, for each positive integer n, an example is given of an entire function of order n having n distinct, prescribed asymptotic functions, each of order less than 1/2.
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Communicated by James K. Langley.
Dedicated to the memory of Walter K. Hayman, FRS.
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Hinkkanen, A., Miles, J. & Rossi, J. Asymptotic Functions of Entire Functions. Comput. Methods Funct. Theory 21, 619–632 (2021). https://doi.org/10.1007/s40315-021-00396-3
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DOI: https://doi.org/10.1007/s40315-021-00396-3